We first prove the case that \(g\)  dose not contain \((y,z,\tilde{z})\) , then \(y\)  is given by the right-continuous version of \(y_t=E\big [\xi +\int _t^Tg(s)ds|\mathcal {F}_t\big ]\) . Thus, for \(\beta \ge 2\) , the martingale representation theorem for locally square integrable martingale in [1]} implies that there exist unique predictable and \(\mathcal {E}\) -predictable \((z,\tilde{z})\in M^{2,\beta }[0,T]\times \tilde{F}^{2,\beta }[0,T]\)  satisfying:
\(E\bigg [\xi +\int _0^Tg(s)ds\bigg |\mathcal {F}_t\bigg ]=y_0+\int _0^tz_sdW_s+\int _0^t\int _{\mathcal {E}}\tilde{z}_{s,e}\tilde{N}(de,ds),\ P\mbox{-}a.s.\) 
